Final answer:
Elvis Presley became uniquely famous for his swinging hips and shaking legs, a move that epitomized the youthful rebellion of rock 'n' roll and contributed to his stardom despite the era's social controversy.
Step-by-step explanation:
The "move" that made Elvis Presley unique and famous early on was his distinctive performance style which included his sultry looks, swinging hips, and shaking legs in time to the music. This move was iconic and represented a form of youthful rebellion that was part of the emerging rock 'n' roll culture. Presley's ability to absorb and blend various regional styles such as gospel, blues, bluegrass, country, western, and R&B shaped his music and contributed to his fame. Despite the controversy his move and rock 'n' roll music generated, particularly among parents who feared it threatened American values, Elvis's popularity soared, especially after network television appearances like those on The Milton Berle Show and The Ed Sullivan Show, where his influential performances triggered both admiration and backlash.